integration - An integral involving error functions and a Gaussian

Let $d\ge 1$ be an integer and let $\vec{A}:=\left\{ A_i \right\}_{i=1}^d$ be real numbers. We consider a following integral:
$$\begin{equation} {\mathfrak I}^{(d)}(\vec{A}):=\int\limits_0^\infty e^{-u^2}\left[ \prod_{i=1}^d \operatorname{erf}(A_i u) \right] du \end{equation}$$
By expanding the error functions in Taylor series and then integrating term by term we found the answer for $d=1$ and $d=2$. We have:

$$\begin{eqnarray} \sqrt{\pi} {\mathfrak I}^{(d)}(\vec{A}) = \begin{cases} \arctan(A_1) & \text{if $d=1$}\\[4pt] \arctan\left(\frac{A_1 A_2}{\sqrt{1+A_1^2+A_2^2}}\right) & \text{if $d=2$} \end{cases} \end{eqnarray}$$
Now the question is how do we derive the result for arbitrary values of $d$?